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<h1 class="title toc-ignore">Compare smashgen and anscombe transform</h1>
<h4 class="author"><em>Dongyue Xie</em></h4>
<h4 class="date"><em>2018-10-13</em></h4>
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<p><strong>Last updated:</strong> 2018-10-18</p>
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<p></details></p>
<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Compare 1. smashgen-ash.identity 2. smashgen-ash.log 3. smashgen-ash.identity.zero 4. smashgen-ash.log.zero (known and unknown nugget effect) with smash-anscombe.</p>
<p>Settings: spike mean function, mean function range (0.1,6) and (20,50)</p>
<p>Note:</p>
<ol style="list-style-type: decimal">
<li>idk: expand around ash posterior mean using identity link in lik_pois; given nugget effect(use <code>sigma=sqrt(nugget^2+s^2)</code> in <code>smash.gaus</code>).</li>
<li>idu: same as 1; unknown nugget effect(use <code>sigma=NULL</code> in <code>smash.gaus</code>).</li>
<li>id0k: expand around ash posterior mean using identity link only for 0 <span class="math inline">\(x\)</span>s and around <span class="math inline">\(x\)</span> for nonzero <span class="math inline">\(x\)</span>s; given nugget effect.</li>
<li>id0u: same as 3; unkown nugget effect</li>
<li>logk: expand around ash posterior mean using log link in lik_pois; given nugget effect</li>
<li>logu: same as 5; unkown nuggect effect</li>
<li>log0k: expand around ash posterior mean using log link only for 0 <span class="math inline">\(x\)</span>s and around <span class="math inline">\(x\)</span> for nonzero <span class="math inline">\(x\)</span>s; given nugget effect.</li>
<li>logu: same as 7; unkown nugget effect</li>
<li>ans: use anscombe transormation on poisson data, variance around 1/4; use 0 variance for 0 <span class="math inline">\(x\)</span>s.</li>
</ol>
<pre class="r"><code>vst_smooth=function(x,method,ep=1e-5){
n=length(x)
if(method=='sr'){
x.t=sqrt(x)
x.var=rep(1/4,n)
x.var[x==0]=0
mu.hat=(smashr::smash.gaus(x.t,sigma=sqrt(x.var)))^2
}
if(method=='anscombe'){
x.t=sqrt(x+3/8)
x.var=rep(1/4,n)
x.var[x==0]=0
mu.hat=(smashr::smash.gaus(x.t,sigma=sqrt(x.var)))^2-3/8
}
if(method=='log'){
x.t=x
x.t[x==0]=ep
x.var=1/x.t
x.t=log(x.t)
mu.hat=exp(smashr::smash.gaus(x.t,sigma=sqrt(x.var)))
}
return(mu.hat)
}
smash_gen_all=function(x,sigma,method){
n=length(x)
if(method=='identity'){
x.ash=ash(rep(0,n),1,lik=lik_pois(x,link='identity'))$result$PosteriorMean
}
if(method=='log'){
x.ash=ash(rep(0,n),1,lik=lik_pois(x,link='log'))$result$PosteriorMean
}
if(method=='identity.zero'){
x.ash=ash(rep(0,n),1,lik=lik_pois(x,link='identity'))$result$PosteriorMean
x.ash[x!=0]=x[x!=0]
}
if(method=='log.zero'){
x.ash=ash(rep(0,n),1,lik=lik_pois(x,link='log'))$result$PosteriorMean
x.ash[x!=0]=x[x!=0]
}
y=log(x.ash)+(x-x.ash)/x.ash
s2=1/x.ash
mu.sigk=exp(smash.gaus(y,sigma=sqrt(sigma^2+s2)))
mu.sigu=exp(smash.gaus(y))
return(list(mu.sigk=mu.sigk,mu.sigu=mu.sigu))
}
simu_study=function(m,sigma=0,nsimu=100,seed=12345){
set.seed(12345)
idk=c()
idu=c()
id0k=c()
id0u=c()
logk=c()
logu=c()
log0k=c()
log0u=c()
ans=c()
for (i in 1:nsimu) {
lambda=exp(log(m)+rnorm(n,0,sigma))
x=rpois(n,lambda)
id=smash_gen_all(x,sigma,'identity')
id0=smash_gen_all(x,sigma,'identity.zero')
logg=smash_gen_all(x,sigma,'log')
log0=smash_gen_all(x,sigma,'log.zero')
idk=rbind(idk,id$mu.sigk)
idu=rbind(idu,id$mu.sigu)
id0k=rbind(id0k,id0$mu.sigk)
id0u=rbind(id0u,id0$mu.sigu)
logk=rbind(logk,logg$mu.sigk)
logu=rbind(logu,logg$mu.sigu)
log0k=rbind(log0k,log0$mu.sigk)
log0u=rbind(log0u,log0$mu.sigu)
ans=rbind(ans,vst_smooth(x,'anscombe'))
}
return(list(idk=idk,idu=idu,id0k=id0k,id0u=id0u,logk=logk,logu=logu,log0k=log0k,log0u=log0u,ans=ans))
}</code></pre>
<p>First we compare all the methods mentioned above using spike mean function whose mean range is around (0.1,6) so there are a number of zero counts in the sequence. This would be a challenge for smashgen since we are using log transformation.</p>
<pre class="r"><code>library(ashr)
library(smashr)
spike.f = function(x) (0.75 * exp(-500 * (x - 0.23)^2) + 1.5 * exp(-2000 * (x - 0.33)^2) + 3 * exp(-8000 * (x - 0.47)^2) + 2.25 * exp(-16000 *
(x - 0.69)^2) + 0.5 * exp(-32000 * (x - 0.83)^2))
n = 512
t = 1:n/n
m = spike.f(t)
m=m*2+0.1
range(m)</code></pre>
<pre><code>[1] 0.100000 6.076316</code></pre>
<pre class="r"><code>result=simu_study(m,sigma=0,nsimu = 4)
mses=lapply(result, function(x){apply(x, 1, function(y){mean((y-m)^2)})})
unlist(lapply(mses, mean))</code></pre>
<pre><code> idk idu id0k id0u logk
5.369863e+02 2.095411e+02 2.827456e-01 1.419588e-01 3.213241e+47
logu log0k log0u ans
2.343187e+15 1.163829e+00 4.926503e-01 7.836854e-02 </code></pre>
<pre class="r"><code>boxplot(mses[-c(1,2,5,6)],main='nugget=0',ylab='MSE')</code></pre>
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<p></details></p>
<p>When there is no nugget effect, id0k, id0u, log0u have relatively smaller mean square error(mse) while anscombe transformation outperforms all smashgen methods and achieves smaller mse.</p>
<p>We plot the estimated mean function of id0u, log0u and ans for comparison. log0u seems to result in underestimations of mean function. id0u overestimates small means and underestimates large means. So when there are a number of zero observations, it’s very crucial to choose where to expand for 0 <span class="math inline">\(x\)</span>s.</p>
<pre class="r"><code>par(mfrow=c(2,2))
for (j in c(1,2,3,4)) {
plot(m,type='l',main='nugget=0')
lines(result$id0u[j,],col=2)
lines(result$log0u[j,],col=3)
lines(result$ans[j,],col=4)
legend('topleft',c('mean','ash identity link','ash log link','anscombe'),lty=c(1,1,1,1),col=c(1,2,3,4))
}</code></pre>
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<p></details></p>
<p>Now we increase nugget effect to <span class="math inline">\(\sigma=1\)</span>. Obviously, using anscombe transformation, we are estimating <span class="math inline">\(exp(\log(\mu)+N(0,\sigma^2))\)</span> so its mse is large and gives spiky fit.</p>
<pre class="r"><code>result=simu_study(m,sigma=1,nsimu = 4)
mses=lapply(result, function(x){apply(x, 1, function(y){mean((y-m)^2)})})
unlist(lapply(mses, mean))</code></pre>
<pre><code> idk idu id0k id0u logk
2.773795e+12 4.929459e+10 5.789335e-01 3.372019e-01 5.332768e+45
logu log0k log0u ans
6.281977e+09 1.168559e+00 6.925573e-01 1.955617e+00 </code></pre>
<pre class="r"><code>boxplot(mses[-c(1,2,5,6)],main='nugget=1',ylab='MSE')</code></pre>
<p><img src="figure/compares.Rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
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<pre class="r"><code>boxplot(mses[-c(1,2,5,6,9)],main='nugget=1',ylab='MSE')</code></pre>
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<p></details></p>
<pre class="r"><code>par(mfrow=c(2,2))
for (j in c(1,2,3,4)) {
plot(m,type='l',main='nugget=1')
lines(result$id0u[j,],col=2)
lines(result$log0u[j,],col=3)
lines(result$ans[j,],col=4)
legend('topleft',c('mean','ash identity link','ash log link','anscombe'),lty=c(1,1,1,1),col=c(1,2,3,4))
}</code></pre>
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<p></details></p>
<p>How about a larger mean function? Increase the range to (20,50). Some observations from the plot: 1. Now, known nugget effect gives smaller mse than unkown ones(e.g idk<span class="math inline">\(<\)</span>idu, id0k<span class="math inline">\(<\)</span>id0u,…); 2. Using identity link in lik_pois is still better; 3. ans has smaller mse but from the plots below, idk, logk and ans give very similar estiamtions.</p>
<pre class="r"><code>m=m*5+20
range(m)</code></pre>
<pre><code>[1] 20.50000 50.38158</code></pre>
<pre class="r"><code>result=simu_study(m,sigma=0,nsimu = 4)
mses=lapply(result, function(x){apply(x, 1, function(y){mean((y-m)^2)})})
unlist(lapply(mses, mean))</code></pre>
<pre><code> idk idu id0k id0u logk logu log0k log0u
2.903710 2.899550 3.317807 3.663308 3.054792 3.054326 3.317807 3.663308
ans
2.815178 </code></pre>
<pre class="r"><code>boxplot(mses,main='nugget=0',ylab='MSE')</code></pre>
<p><img src="figure/compares.Rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /></p>
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<p></details></p>
<p>Plots compare idk, logk and ans:</p>
<pre class="r"><code>par(mfrow=c(2,2))
for (j in c(1,2,3,4)) {
plot(m,type='l',main='nugget=0')
lines(result$idk[j,],col=2)
lines(result$logk[j,],col=3)
lines(result$ans[j,],col=4)
legend('topleft',c('mean','ash identity link','ash log link','anscombe'),lty=c(1,1,1,1),col=c(1,2,3,4))
}</code></pre>
<p><img src="figure/compares.Rmd/unnamed-chunk-7-1.png" width="1152" style="display: block; margin: auto;" /></p>
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<p></details></p>
</div>
<div id="summary" class="section level1">
<h1>Summary</h1>
<p>Maybe can develop a version of anscombe to deal with nugget effect? Also is nugget effect necessarily defined as <span class="math inline">\(exp(\log(\mu)+\sigma^2)\)</span>?</p>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] smashr_1.2-0 ashr_2.2-7
loaded via a namespace (and not attached):
[1] Rcpp_0.12.18 knitr_1.20 whisker_0.3-2
[4] magrittr_1.5 workflowr_1.1.1 REBayes_1.3
[7] MASS_7.3-50 pscl_1.5.2 doParallel_1.0.14
[10] SQUAREM_2017.10-1 lattice_0.20-35 foreach_1.4.4
[13] stringr_1.3.1 caTools_1.17.1.1 tools_3.5.1
[16] parallel_3.5.1 grid_3.5.1 data.table_1.11.6
[19] R.oo_1.22.0 git2r_0.23.0 htmltools_0.3.6
[22] iterators_1.0.10 assertthat_0.2.0 yaml_2.2.0
[25] rprojroot_1.3-2 digest_0.6.17 Matrix_1.2-14
[28] bitops_1.0-6 codetools_0.2-15 R.utils_2.7.0
[31] evaluate_0.11 rmarkdown_1.10 wavethresh_4.6.8
[34] stringi_1.2.4 compiler_3.5.1 Rmosek_8.0.69
[37] backports_1.1.2 R.methodsS3_1.7.1 truncnorm_1.0-8 </code></pre>
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